Sufficient conditions for total positivity, compounds, and Dodgson condensation

TL;DR

This paper investigates the conditions under which compounds and Dodgson condensations of totally positive matrices retain total positivity, revealing negative results generally but positive results for Hankel matrices.

Contribution

It provides new insights into the preservation of total positivity under compounds and Dodgson's condensation, especially for Hankel matrices.

Findings

Compounds of general TP matrices are not necessarily TP.

Dodgson's condensation does not generally preserve TP.

Condensed matrices of TP Hankel matrices are always TP.

Abstract

A $n$-by-$n$ matrix is called totally positive ($TP$) if all its minors are positive and $TP_k$ if all of its $k$-by-$k$ submatrices are $TP$. For an arbitrary totally positive matrix or $TP_k$ matrix, we investigate if the $r$th compound ($1<r<n$) is in turn $TP$ or $TP_k$, and demonstrate a strong negative resolution in general. Focus is then shifted to Dodgson's algorithm for calculating the determinant of a generic matrix, and we analyze whether the associated condensed matrices are possibly totally positive or $TP_k$. We also show that all condensed matrices associated with a $TP$ Hankel matrix are $TP$.